Evolution gave humans advanced grasping capabilities combining an adaptive hand with efficient control. Grasping motions can quickly be adapted if the object moves or deforms. Soft-grasping with an anthropomorphic hand is a great capability for robots interacting with objects shaped for humans. Nevertheless, most robotic applications use vacuum, 2-finger or custom made grippers. We present a biologically inspired spiking neural network (SNN) for soft-grasping to control a robotic hand. Two control loops are combined, one from motor primitives and one from a compliant controller activated by a reflex. The finger primitives represent synergies between joints and hand primitives represent different affordances. Contact is detected with a mechanism based on inter-neuron circuits in the spinal cord to trigger reflexes. A Schunk SVH 5-finger hand was used to grasp objects with different shapes, stiffness and sizes. The SNN adapted the grasping motions without knowing the exact properties of the objects. The compliant controller with online learning proved to be sensitive, allowing even the grasping of balloons. In contrast to deep learning approaches, our SNN requires one example of each grasping motion to train the primitives. Computation of the inverse kinematics or complex contact point planning is not required. This approach simplifies the control and can be used on different robots providing similar adaptive features as a human hand. A physical imitation of a biological system implemented completely with SNN and a robotic hand can provide new insights into grasping mechanisms.